On the Hardy number and the Bergman number of a planar domain

Abstract: This article deals with functions with a prefixed range and their inclusions in Hardy and weighted Bergman spaces. This idea was originally introduced by Hansen for Hardy spaces, and it was recently taken into weighted Bergman spaces by Karafyllia and Karamanlis. We provide a new characterization for the Hardy number of a domain in terms of its Green function. Based on this, we present a class of domains for which the Hardy number and the Bergman number coincide. However, in general, we show that the Hardy number and the Bergman number of a domain are not equal; even for domains which are regular for the Dirichlet problem.